VoL  »T,  No.  «6.  BULLETi:N\MAS§*eHUSETTS  INSTITUTE  OF  TECHNOLOGY,  May,  19M 

Entered  December  3,  1904,  at  the  Post-Office,  Boston,  Mass.,  as  second-class  matter, 

under  act  of  Congress  of  July  1C,  1894 

Pub.  Serial  No.  87 


PUBLICATIONS 

OF   THE 

Massachusetts 

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Institute  of  Technology 


Contribution  from 
Division  of  Industrial  Co-operation^and  Research 


Serial  No.  1 


May,  1922 


PHOTO-ELASTICITY  AND  ITS  APPLICATIONS 
TO  ENGINEERING   PROBLEMS 

By 
Paul  Heymans 


Alls; 


PHOTO-ELASTICITY 

AND  ITS  APPLICATION  TO 

ENGINEERING  PROBLEMS 


By 
PAUL  HEYMANS,  D.Eng.Sc. 

University  of  Ghent  (Belgium) 


REPRINTED  FROM 

THE  TECH  ENGINEERING  NEWS 

ISSUE  OF  JUNE.  1922 


80 


THE  TECH  ENGINEERING  NEWS 


June,  1922 


PHOTO-ELASTICITY  AND  ITS  APPLICATION  TO 

ENGINEERING  PROBLEMS 

By  PAUL  HEYMANS.  D.  Eng.  Sc.,  University  of  Ghent  (Belgium) 
Research  Associate,  Research  Laboratory  of  Industrial  Physics.  Massachusetts  Institute  of  Technology 

(Copyright,  192$,  by  Division  of  Industrial  Co-operation  and  Research,  Massachusetts  Institute  of  Technology.     All  rights  reserved  including  translation  into  foreign  languages, 

including  the  Scandinavian.) 


One  of  the  important  and  often 
difficult  problems  with  which  an  engi- 
neer has  to  deal  is  the  determination 
of  the  shape  and  the  dimensions  which 


FIG.  i 

DR.  COKER'S   LATERAL  EXTENSOMETER 

he  must  give  to  the  different  parts  of  a 
proposed  construction.  Several  factors 
have  to  be  considered,  but  the  domi- 
nant ones  are  safety  and  economy. 


As  we  know,  materials  yield  at 
points  where  the  stress  exceeds  the 
elastic  limit.  If  we  were  able  to  make 
a  complete  determination  of  what  the 
stress  would  be  at  each  point  of  the 
structure,  it  would  be  easy  to  give  to 
all  parts  adequate  dimensions;  a  min- 
imum of  material  would  be  wasted 
and  safety  would  be  secured. 

Notwithstanding  the  progress  made 
in  mathematical  treatment  of  elastic 
problems,  the  determination  of  the 
stress  distribution  remains  in  most 
cases  an  unsolved  problem.  Where 
theoretical  solutions  exist,  even  in  the 
relatively  simple  cases,  they  usually 
lead  to  lengthy  and  intricate  calcula- 
tions, and  it  cannot  be  expected  that 
even  a  well-trained  engineer  will  use 
them  in  practice. 

The  uncertainty  of  knowledge  of  the 
stress  distributions,  on  which  the 
determination  of  the  dimensions  of 
the  different  parts  of  a  construction 
has  to  be  based,  requires  the  use  of  an 
excessive  factor  of  safety.  All  the 
material  added  above  that  strictly 
necessary  means  waste.  Accidents 
frequently  occur  which  show  that, 
notwithstanding  all  precautions,  safety 
in  design  has  not  been  secured. 

The  object  of  the  mathematical 
theory  of  elasticity  is  the  analytical 
determination  of  this  stress  distribu- 
tion. Photo-Elasticity,  consisting  of 
recently  developed  methods  of  optical 
investigation  of  the  stress  distribution, 


Polarizer       Model  tinder  examination        Comparison  member    Analyzer       White  screen 


leads  to  an  experimental  solution  for 
all  two-dimensional  elastic  problems, 
provided  the  material  used  is  isotropic 
and  obeys  Hooke's  law  of  linear  pro- 
portionality between  stress  and  strain, 
i.e.  within  the  elastic  limit  of  the 
material. 

As  we  know,  the  number  of  cases,  for 
which  a  complete  mathematical  solu- 
tion for  the  determination  of  stress 


GENERAL  VIEW  OF  APPARATUS  USED  IN  THE  OPTICAL  STRESS  ANALYSIS 


FIG.  4 
LINES   OF   PRINCIPAL  STRESS 

distribution  exists,  is  limited,  and  the 
calculations  worked  out  in  practice  for 
the  computation  of  stability  are,  in 
general,  an  acaptation  of  the  ideal  or 
incomplete  theoretical  solutions  to  the 
needs  of  engineering  practice.  The 
simplifying  assumptions,  which  have 
to  be  made,  may  be  seen,  by  closer 
analysis  and  especially  by  photo- 
elastic  investigation,  to  be  quite  often 
of  doubtful  accuracy,  and  may  lead, 
especially  in  new  or  unusual  problems, 
to  dangerous  approximations. 

I.   GENERAL  PRINCIPLES  OF 
ELASTICITY  AND  PHOTO- 
ELASTICITY 

The  state  of  stress  at  any  point  of  a 
body  is  determined  when  the  traction 
across  every  plane  through  the  point 
is  known.  There  exist  at  any  {joint 
three  orthogonal  planes,  across  which 
the  traction  is  purely  normal  and 
which  are  called  the  planes  of  principal 
stress.  The  normal  tractions  across 
those  planes  are  called  the  principal 
stresses.  So  the  state  of  stress  at  any 
point  is  completely  determined  by  the 
direction  and  the  magnitude  of  the 
principal  stresses  at  the  point  under 
examination.  These  are  therefore  the 
elements  which  we  determine  in  a 
stress  analysis. 

Most  of  the  isotropic  transparent 
bodies,  such  as  glass,  or  still  better 
such  as  xylonite  (1)  are  optically 
inactive  in  their  normal  state,  but 
show  double  refraction  when  put 
under  stress.  The  photo-elastic  methods 
of  stress  investigation,  whose  develop- 

(1)  Xvlonite  is  camphorated  nitro-cellulose,  of  the 
same  kind  as  celluloid. 


June,  1922 


FIG.  i 

BEAM   OF   RECTANGULAR   CROSS  SECTION 
UNDER   LONGITUDINAL  PULL 

merit,  as  described  in  this  article, 
has  been  mostly  carried  out  under 
the  direction  of  Prof.  E.  G.  Coker 
(University  College,  University  of 
London),  are  based  upon  these  tem- 
porary birefracting  properties  shoirn  by 
transparent  bodies  ichen  stressed. 

If  plane  polarized  light  is  first  passed 
through  a  stressed  specimen  of  xylonite 
and  afterwards  through  a  second  Xicol 
prism,  whose  principal  section  is  par- 
allel to  the  plane  of  polarization  of  the 
original  beam  of  light,  only  the  points 
where  the  principal  stresses  are  re- 
spectively parallel  and  perpendicular 
to  the  principal  sections  of  the  crossed 
Xicols  remain  dark.  This  property 
enables  us  to  determine  the  directions  cj 
the  principal  stresses  at  any  given  point. 

If  now  we  pass  through  the  specimen 
circularly  polarized  light,  by  interfer- 
ence of  the  two  component  rays  (which 
in  the  double-refracting  specimen  have 
suffered  a  relative  retardation,  depend- 
ing at  each  point  on  the  magnitude  of 
the  two  principal  stresses),  a  colored 
image  is  obtained. 

By  a  comparison  method,  based 
upon  the  interposition  in  a  suitable 
direction  of  a  comparison  member  of 
constant  cross  section,  put  under 
uniform  tension  in  an  appropriate 
frame,  we  read  on  the  dynamometer  of 
the  frame  the  value  of  the  difference  of 
the  principal  stresses  at  any  given  point. 

Now,  in  the  two  dimensional  elastic 
problems,  the  transverse  deformation, 


THE  TECH  ENGINEERING  NEWS 

i.e.,  the  deformation  along  a  normal  to 
the  plane  of  the  two  principal  stresses, 
is  proportional  to  the  sum  of  those  two 
stresses.  By  means  of  Dr.  Coker's 
lateral  extensometer  (Fig.  1)  we  mea- 
sure this  transverse  deformation. 

From  the  ralues  of  the  differences  and 
the  sums  of  the  principal  stresses,  ice 
compute  the  separate  tallies  of  each  of 
them,  so  determining  completely  the 
elastic  state. 

The  question  naturally  occurring  to 
an  engineer  is  whether  the  results, 
obtained  on  a  transparent  body  such  as 
xylonite,  are  of  any  value  for  engineer- 
ing materials.  It  is  shown  by  the  gen- 
eral discussion  of  the  equations  of 
elastic  equilibrium  that  in  the  case 
either  of  plane  strain  or  of  plane  stress, 
in  an  isotropic  body,  obeying  Hooke's 
law  of  linear  proportionality  between 
strain  and  stress,  the  stress  distribution 
is  independent  of  the  moduli  of  elas- 
ticity and  consequently  of  the  nature  of 
the  body.  So,  the  stress  distribution 
experimentally  determined  on  xylonite 
is  the  same  in  any  other  isotropic  sub- 
stance obeying  Hooke's  law,  among 
others,  in  steel  (2).  Moreover  these 
conclusions  derived  from  the  general 
theory  of  elasticity  have  been  checked 
by  experimental  work. 

II.     PHOTO-ELASTIC  STRESS 
ANALYSIS  APPLIED  TO: 

A.     Some  simple  cases. 

Let  us  first  consider  the  application 
of  photo-elasticity  to  a  few  simple 
problems  in  order  to  make  clear  the 
different  processes. 

(4)  Except,  however,  if  the  body  is  multiply-connected 
and  the  resultant  applied  forces  do  not  vanish  separately 
over  each  boundary.  In  this  particular  case  the  correction 
coefficients  for  passing  from  one  isotropic  substance  to 
another,  having  different  values  for  the  elastic  constants, 
may  be  experimentally  determined.  ("On  Stresses  in 
Multiply-Connected  Plates."  by  L.  X.  G.  Filon.  British 
Assoc.  Report,  1941.) 


81 


1.  If  we  pull  uniformly  a  xylonite 
beam  of  rectangular  cross  section,  which 
is  optically  inactive  in  its  normal  state, 
the  uniform  color  which  will  appear,  as 


:  FIG.  s 

THE  PRINCIPAL  STRESSES   ACROSS  THE 

MINIMUM   CROSS  SECTION   AND  ALONG   THE 

OUTSIDE   EDGE 

it  is  put  under  stress,  shows  us  (Fig.  2) 
where  we  have  uniformly  distributed 
tension. 

If,  in  the  central  part  of  this  same 
beam,  previously  under  uniform  ten- 
sion, a  circular  hole  is  drilled,  free  from 
initial  machining  stresses,  the  image 
r.utained  (Fig.  3)  shows  that  this 
internal  discontinuity  causes  a  very 
different  distribution  of  stress.  For 
one  not  familiar  with  these  phenomena, 
the  interpretation  of  such  images  may 
be  somewhat  difficult.  It  may  be  made 
easier  by  pointing  out  that  the  iso- 
chromatic  lines  or  zones  (lines  or 
zones  of  same  color)  correspond  to 
equal  values  of  difference  of  principal 


FIG.  s 


A  CIRCULAR  HOLE  IN  A  BEAM  OF  RECTANGULAR  CROSS  SECTION  UNDER  LONGITUDINAL  PULL 


M118992 


82 


THE  TECH  ENGINEERING  NEWS 


June,   1922 


FIG.  8 

Stresses  at  the  Ends  of  a  Crack  „. „  „.  ,.,„.  ,,,„„„  „,  [ne  ^racl 

A   CJIACK  IN  A  BEAM  OF  RECTANGULAR  CROSS  SECTION  UNDER  LONGITUDINAL  PULL 


FIG.  9 
A  Circular  Hole  Drilled  at  the  Ends  of  the  Crack 


stresses.  The  image  shows,  among 
other  things,  that  the  stress  is  not 
uniform  across  the  minimum  cross 
section  and  that  therefore  the  engineer 
makes  an  approximation,  already  in 
this  very  simple  case,  when  he  bases 
the  calculation  of  the  cross  section  on 
the  mean  stress  through  the  active 
section. 

Tne  analysis  of  the  stress  distribu- 
tion in  a  beam  of  rectangular  section, 
in  which  a  circular  hole  has  been 
drilled  (3),  is  shown  on  Figs.  4,  5  and  6. 

Fig.  4  shows  what  are  called  the 
lines  of  principal  stress.  The  tangent 
and  the  normal  to  those  lines  define 
at  each  of  their  points  the  directions 
of  the  principal  stresses. 
L  Fig.  5  shows  the  values  of  the  prin- 
cipal stresses  across  the  minimum 
cross  section,  and  the  polar  diagram 
of  Fig.  6  shows,  with  the  boundary  of 


FIG.  a 

THE  (TANGENTIAL  STRESS  ALONG   THE 
BOUNDARY  OF  THE  CIRCULAR   HOLE 

the  hole  as  origin  for  the  values  of  the 
stresses,  the  value  of  the  tangential 
stress  at  the  boundary  of  the  circular 
discontinuity. 

(S)  Diagrams  4,  5  and  6  are  taken  from:  "The  effects  of 
holes  and  semi-circular  notches  on  the  distribution  of 
stress  in  tension  members,"  by  E.  G.  Coker  (Proc.  Physical 
Soc.  London,  vol.  XXV,  Part  II). 


What  is  most  striking  here  is: 
For  the  diagram  of  Fig.  5,  that 
the  maximum  value  of  the  stress, 
which  occurs  at  the  inside  edge,  is 
equal  to  about  1.8  times  the  mean 
stress.  That  ratio  is  the  approxima- 
tion made  by  the  engineer  when  he 
calculates  the  section  assuming  that 
the  stress  is  uniformly  distributed. 

For  the  diagram  of  Fig.  6,  that 
the  stress  at  the  inside  points  of  the 
hole  where  the  longitudinal  axis  of 
the  member  cuts  the  boundary  of  the 


hole,  the  tangential  stress  is  a  com- 
pression. 

2.  If  in  a  similar  beam  of  rectangu- 
lar section,  we  cut  an  elliptical  hole, 
whose  major  axis  is  perpendicular  to 
the  direction  of  the  pull,  the  stress  is, 
as  in  the  previous  case,  concentrated 
near  the  boundary  of  the  disconti- 
nuity; the  maximum  value  of  the  stress 
occurs  at  the  point  of  the  boundary 
where  the  tangent  to  the  ellipse  is 
parallel  to  the  direction  of  the  pull. 

The  values  of  the  stress  at  the 
boundary  of  the  hole  are  given  (4)  by 
Fig.  7,  (p.  84)  where  we  see  that  the 
maximum  stress  is  equal  to  four  times 
the  mean  stress  and  that  the  compres- 
sion at  the  end  of  the  minor  axis  (90°) 
of  the  elliptical  hole  is  equal  to  the 
mean  stress. 

The  value  of  the  maximum  stress, 
at  the  point  where  the  tangent  to  the 
ellipse  is  parallel  to  the  direction  of 
pull  (0°),  is,  as  has  been  mathemat- 
ically shown  (5),  a  function  of  the  ratio 
of  the  lengths  of  the  two  axes.  If  we 
admit  that  a  crack  may  be  considered 
as  an  elliptical  discontinuity  such  that 
one  of  the  axes  becomes  very  small,  the 
fact  that  a  crack  usually  keeps  on 
extending,  even  if  the  member  is  only 
slightly  stressed,  is  explained  by  this 
high  stress  concentration  due  to  the 

(4)  Diagram  7  is  taken  from:  "The  effects  of  holes, 
cracks  and  other  discontinuities  in  ships  plating,"  by 
E.  G.  Coker  aad  A.  L  Kimball,  Jr.  (Trans.  Inst.  Naval 
Architects,  London,  19^0.) 

(.5)  "Stresses  in  a  plate  due  to  the  presence  of  cracks 
and  sharp  corners,"  by  C.  E.  Inglis.  (Trans.  Inst.  Naval 
Architects,  London,  1913.) 


FRAMES   USED  FOR  STRESSING  SMALL  MODELS 


June,   1922 


THE  TECH  ENGINEERING  NEWS 


83 


high  value  of  the  ratio  of  the  major 
axis  to  the  minor.  Fig.  8  shows  the 
high  concentration  of  stress  occurring 
at  the  end  of  such  a  crack;  and  the 
way,  well  known  to  engineers,  to  limit 
the  extension  of  a  crack,  which  is 
starting,  is  to  replace  the  ends  of  the 
elliptical  discontinuity  by  a  circular 


section,  and  Fig.  14  (p.  84)  the  values  of 
the  tangential  stress  along  the  semi- 
circular notch  and  along  the  outside 
parallel  edge.  Again  here  the  value 
of  the  maximum  principal  stress  is 
about  30%  higher  than  the  mean 
stress  through  the  minimum  cross  sec- 
tion. Figs.  15  to  20  inc.  (p.  85  and  96) 


FIG.  il 

FOUR  PARALLEL  SCRATCHES  AT  THE 
SURFACE  OF  A  BENT  BEAM 

one,  where  the  ratio  of  the  axes  is 
equal  to  one:  by  drilling  two  circular 
holes,  we  get  an  image  as  shown  by 
Fig.  9,  corresponding  to  a  more  uni- 
form distribution  of  the  stresses  and 
consequently  to  smaller  stress  con- 
centration. The  replacement  of  the 
elliptical  crack  by  two  elliptical  holes 
whose  axes  are  normal  to  those  of  the 
crack  has  been  shown  to  be  still  more 
advantageous. 

Isodinic  Lines 


FIG.  45 
A   XYLONITE  MODEL  OF  A  GEAR   WHEEL 


give  the  values  of  the  principal  stresses 
(7)  across  the  minimum  cross  section 
and  along  the  edge  of  the  lateral  discon- 
tinuity, respectively  for  the  V-notch, 
for  the  U-notch  and  for  the  Charpy 
notch  (impact  tests),  in  a  beam  of 
rectangular  cross  section  under  longi- 
tudinal pull. 

(7)  Diagrams  15  to  40  are  taken  from:  "Stress  concen- 
trations due  to  notches  and  like  discontinuities."  by  E.  G. 
Coker  and  Paul  Heymans,  (British  Assoc,  Report.  1941) 
and  "  Etude  par  la  photo-ela>t  icimet  rie  de  la  distribution  des 
surtensions  dues  a  certaines  discontinuites  dans  les  pieces 
•oaillillfs  a  traction."  by  Paul  Heymans.  (Academic  Royale 
de  Belgique,  Classe  des  Sciences.  Brussels.  19*1.) 


The  maximum  stress  depends  on  the 
radii  of  the  curves  at  the  bottom  of 
the  notch.  If  for  the  V-notch  this 
radius  becomes  small,  we  have  a 
scratch,  as  those  shown  by  Fig.  21, 
which  represents  the  effect  of  four 
parallel  scratches  at  the  surface  of  a 
bent  beam.  The  higher  maximum 
stress  which  is  revealed  by  this  photo- 
elastic  analysis  shows  why  a  scratch, 
which  is  surely  not  an  appreciable 
reduction  of  the  active  section,  weakens 
considerably  a  member,  especially  if  it 
is  made  of  brittle  material,  such  as 
glass  or  hardened  steel,  in  which 
practically  no  redistribution  of  stress 
occurs  between  the  elastic  limit  and 
the  breaking  point. 

B.  Some  specific  engineering  problems 
A  great  number  of  problems  occur- 
ring in  structural  engineering,  such  as 
bridge  construction,  design  of  trans- 
mission towers,  naval  architecture,  etc., 
either  are  or  may  be  decomposed  into 
two-dimensional  elastic  problems,  and 
therefore,  however  hyperstatic  and 
indeterminate  they  may  be,  are  cap- 
able of  being  solved,  as  completely  as 
desired,  by  the  photo-elastic  methods. 
1.  Fig.  22  (p.  96)  for  instance  shows 
the  model  of  a  simple  truss,  which  has 
been  used  for  experimental  purposes 
(Prof.  Coker's  laboratory)  under  differ- 
ent combinations  of  applied  loads.  The 
stress  analysis  at  the  different  joints 
i£  of  course  the  most  interesting.  The 
image  obtained  by  photo-elastic  anal- 
ysis (8)  shows  immediately  that  the 

(8)  "  La  Photo-elasticimetrie,  ses  principes.  ses  met  nodes 
et  ses  applications,"  by  Paul  Heymans,  (Bull  Soc.  Beige 
Ing.  et  Ind.,  Brussels,  19*1).  p.  IIS. 


FIG.   14.     Lines  of  Principal  Stress 

3.  If  again  in  a  similar  beam,  we 
cut  out  at  both  sides  two  symmetrical 
semi-circular  notches,  we  get  the  image 
shown  by  Fig.  10. 

A  V-notch  (Fig.  11)  shows  a  some- 
what similar  stress  distribution.  For 
the  semi-circular  notch,  the  lines  of 
principal  stress  (6)  are  shown  by  Fig. 
12.  Fig.  13  (p.  84)  shows  the  values  of 
the  principal  stresses  for  a  notch  of  % 
inch  radius  across  the  minimum  cross 

(6)  Diagrams  14,  13  and  14  are  taken  from:  "Pboto- 
elastic  measurements  of  the  stress  distribution  in  tension 
members  used  in  the  testing  of  materials,"  by  E.  G.  Coker. 
Proc.  Inst.  Civil  Eng..  London,  1941.) 


FIG.   10  FIG.  11 

Lateral  Semi-Circular  Notch  Lateral  V-Shaped  Notch 

LATERAL  NOTCHES  IN  A  BEAM  OF  RECTANGULAR  CROSS  SECTION  UNDER  LONGITUDINAL  PULL 


84 


THE  TECH  ENGINEERING  NEWS 


June,  1922 


amount  of  calculations  made  was  con- 
siderable, and,  while  checking  them, 
certain  errors  in  the  methods  and  in 
the  figures  had  been  detected.  Com- 
plementary calculations,  sent  in  as 
corrections  of  the  first  one,  extended 
into  a  book  of  forty-eight  pages.  Were 
there  no  other  errors  which  had  been 


FIG. 


FIG.  28 

A  Xylonite  Modej  under  inside  pressure  due  to  the 
Shrinking  on  the  Shaft 

PHOTO-ELASTIC  STUDY  OF  GEAR  WHEELS  AND  PINIONS 


The  Xylonite  Model  when  the  Torque  is  applied 


members  are  not,  as  assumed  in  calcu- 
lations, under  uniform  stress.  In  fact, 
the  image  shows  the  actual  state  of 
stress  at  each  point,  due  as  well  to  the 
so-called  primary  as  to  the  secondary 
stresses. 

2.  Application  of  photo-elastic  anal- 
ysis has  been  made  successfully  to 
bridge  construction. 

An  interesting  case  is  the  one  made 
and  published  by  M.  Mesnager,  Chief 


of 


Engineer     of     the     Department 
Bridges  and  Roads  in  Paris  (9). 

Fig.  23  (p.  100)  shows  a  general  view 
of  the  bridge  and  Fig.  24  represents 
the  models  and  special  frames  used. 


FIG.  13 

THE   PRINCIPAL  STRESSES  ACROSS   THE 
MINIMUM   CROSS  SECTION 

It  is  interesting  to  quote  M.  Mes- 
nager (10),  "We  have  used  photo- 
elastic  analysis  in  the  laboratory  of 
the  Ecole  des  Fonts  et  Chaussees 


SPECIAL  PHOTO-ELASTIC  TENSION  AND 

COMPRESSION  MACHINE  DESIGNED  BY 

A.  L.  KIM  BALL,  JR. 


FIG.  14 

THE  TANGENTIAL  STRESS  ALONG   THE 

BOUNDARY  OF  THE  NOTCH  AND  ALONG 

THE  OUTSIDE  EDGE 

(Paris)  in  order  to  check  the  calcula- 
tions made  for  a  bridge  of  ninety-five 
meters  span,  which  was  to  be  built 
over  the'  Rhone  at  la  Balme.  .  .  .  The 

(9)  "Utilisation  de  la  double  refraction  accidentelle  du 
verre  a  1'etude  des  efforts  interieurs  dans  les  solides, "  by 
M.   Mesnager.     (Annales   des  Ponts  et  Chaussees,  Paris, 
1913.) 

(10)  Translated  from  M.  Mesnager,  loc.  cit.,  p.  178  and 
p.  135. 


FIG.  7 

THE   TANGENTIAL  STRESS  ALONG   THE 
BOUNDARY   OF  THE   ELLIPTICAL  HOLE 

overlooked?  Besides,  for  such  an 
important  construction  in  reinforced 
concrete,  was  it  not  advisable  to  make 
certain  preliminary  experimental  veri- 
fications? Tests  made  in  the  labora- 
tory on  small  metal  models  had  given 
some  interesting  results,  but  most  of 
them  were  incapable  of  interpretation. 


FIG. 


THE   GLASS  MODEL  AND  THE    FRAMES   USED 
BY  MESNAGER  IN  HIS  PHOTO-ELASTIC 
ANALYSIS   OF  THE   LA   BALME  BRIDGE 


June,  1922 


THE  TECH  ENGINEERING  NEWS 


85 


On  the  other  hand,  reliable  experi- 
mental results  were  asked  by  the 
Acting  Commission  before  decision. 
It  is  in  those  circumstances  that,  in 
order  to  come  to  a  solution,  I  had  a 


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FIG.  15 

THE  PRINCIPAL  STRESSES  ACROSS  THE 
MINIMUM   CROSS  SECTION  (V-Notch) 

glass  model  of  the  bridge  made.  This 
work,  which  cost  less  than  one- 
thousandth  of  the  price  of  the  con- 
struction, finally  enabled  us  to  get 
the  necessary  verifications.  .  .  .  The 


FIG. 


THE  TANGENTIAL  STRESS  ALONG  THE 
BOUNDARY  OF  THE  NOTCH 

study  of  the  reduced  glass  model 
enabled  us  to  obtain  accurate  results 
in  less  than  twenty  days,  whereas 
the  calculus  had  required  a  time  con- 
siderably longer  without  giving  the 
same  reliability." 


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FIG    17 

THE   PRINCIPAL  STRESSES  ACROSS  THE 
MINIMUM   CROSS  SECTION  (U -Notch) 

However  at  the  time  when  this 
application  of  photo-elastic  analysis 
was  made,  only  the  maximum  stresses, 
which  always  occur  at  the  edges  where 
one  of  the  principal  stresses  vanishes, 
could  be  obtained  and,  furthermore, 
Mesnager  used  glass,  which  gave  him 
serious  difficulties  for  the  building  of 
the  models. 


FIG.  18 

THE  ^TANGENTIAL  STRESS  ALONG   THE 
BOUNDARY   OF  THE   NOTCH 

3.  In  the  photo-elastic  laboratory 
here  at  Technology  an  extended  study 
of  stress  distribution  in  certain  special 
gear  teeth  and  pinions  is  now  being 
carried  out. 


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S    H    a    ».-    II      10    3'    g      7      f      i      4      J  '     T 

a 

FIG.  19 

THE  PRINCIPAL  STRESSES  ACROSS  THE 
MINIMUM  CROSS  SECTION  (Charpy  Notch) 

Fig.  25  (p.  83)  shows  one  of  the  xylon- 
ite models  when  no  force  is  applied. 

Fig.  26  (p.  84)  give:-  the  colored  image 
obtained  for  the  structure  in  position 
on  the  shaft. 


FIG.  «o 

THE  TANGENTIAL  STRESS  ALONG  THE 
BOUNDARY  OF  THE  NOTCH 


Fig.  27  (p.  84)  gives  the  colored  image 
when  the  torque  is  applied. 

The  work  being  still  in  progress,  it 
would  be  premature  to  discuss  results. 


When  one  has  examined  the  stress 
distribution  in  a  certain  number  of 
members  entering  into  a  structure,  he 
is  impressed  with  the  number  and 
nature  of  the  approximation  employed 
in  engineering  problems.  Empirical 
calculations  of  that  kind,  —  and  prac- 
tically all  contain  simplifying  assump- 
tions, -  -  may  be  relied  upon  only 
when  they  have  been  sufficiently 
checked  by  experience,  —  if,  in  other 
words,  the  factor  of  safety  has  been 
for  each  particular  case  adequately 
adjusted. 


FIG.  w 

A   XYLONITE   MODEL  OF  A  SIMPLE   TRUSS 

For  new  types  of  construction,  or 
where  it  becomes  of  importance  to 
distribute  the  structural  material  in 
the  most  economical  manner,  these 
approximate  calculations  are  often 
inadequate.  It  may  be,  as  in  airplane 
design,  that  the  purpose  is  to  lighten 
the  structure,  or  as  in  the  designs  of 
steel  and  concrete  constructions,  to 
reduce  the  amount  of  idle  material. 


FIG.  23 
GENERAL    VIEW  OF  THE  LA   BALME   BRIDGE 

By  the  use  of  photo-elastic  methods 
these  objects  can  be  obtained  without 
compromising  safety.  The  constantly 
recurring  accidents  caused  by  failures 
of  mechanical  devices  and  structures, 
emphasize  the  need  of  a  method  of 
analysis  such  as  is  now  offered  through 
the  photo-elastic  investigations. 

The  photo-elastic  method  of  investigation 
has  been  developed  principally  by  Dr.  E.  G. 
Coker  of  University  College,  University  of 
London,  with  whom  the  writer  has  studied  and 
to  whom  he  is  greatly  indebted  for  inspiration 
and  suggestions  concerning  this  later  work. 
The  author  also  acknowledges  the  assistance  of 
Mr.  John  T.  Norton  in  taking  many  photo- 
graphs including  the  autochrome  color  plates, 
used  in  this  article,  and  of  Mr.  Carl  Selig  in  cut- 
ting the  xylonite  models  and  the  General  Electric 
Company  in  loaning  a  portion  of  the  apparatus  . 


14  DAY  USE 


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